We perform a detailed Hopf bifurcation analysis to the model, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.. Th
Trang 1R E S E A R C H Open Access
Bifurcation analysis of a diffusive model of
pioneer and climax species interaction
Jianxin Liu and Junjie Wei*
* Correspondence: weijj@hit.edu.cn
Department of Mathematics,
Harbin Institute of Technology,
Harbin, Heilongjiang 150001, PR
China
Abstract
A diffusive model of pioneer and climax species interaction is considered We perform a detailed Hopf bifurcation analysis to the model, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions
Keywords: pioneer and climax, Hopf bifurcation, diffusive model
1 Introduction
We consider the following model:
u t = d1u + uf (c11u + v),
where xÎ Ω, t > 0, and u, v represent a measure of a pioneer and a climax species, respectively f(z), the growth rate of the pioneer population, is generally assumed to be smoothly deceasing, and has a unique positive root at a value z1 so that the crowding
is particularly harmful for pioneer species But for the climax population, it is different from pioneer population Climax fitness increases at low total density but decreasing at higher densities So that, it has an optimum value of density for growing Hence, g(z), the growth rate of the climax population, is assumed to be non-monotone, has a hump, and possesses two distinct positive roots at some values z2 and z3, with z2 <z3
and g’(z2) > 0 >g’(z3) For the reason above, we set
f (c11u + v) = z1− c11u − v,
in this article
Equation (1.1) is often used to describe forestry models Examples can be found in [1,2] and references therein The dynamics of pioneer-climax models have been studied widely Systems described by ordinary differential equations are under the hypothesis
of homogeneous environment The stability of positive equilibrium and bifurcation, especial Hopf bifurcation are the subject of many investigations More recently, the environmental factors are introduced to the pioneer-climax systems Models including diffusivity (i.e systems described by reaction-diffusion equations) have been considered The existence of positive steady state solutions are the subject of investigations
© 2011 Liu and Wei; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,
Trang 2In addition, traveling wave solutions are the most interesting problem The readers can
get some results from [3] In bifurcation problems, Buchanan [4] has studied Turing
instability in a pioneer/climax population interaction model He determined the values
of the diffusional coefficients for which the model undergoes a Turing bifurcation, and
he show that a Turing bifurcation occurs when an equilibrium solution becomes
unstable to perturbations which are nonhomogeneous in space but remains stable to
spatially homogeneous perturbations Hopf bifurcation for diffusive pioneer-climax
spe-cies interaction has not been studied Our study will be performed in Hopf bifurcation
The rest of this article are structured in the following way: in Section 2, the condi-tions of the existence of positive equilibrium are given The critical values of the
para-meter for Hopf bifurcation occurring are also searched And the stability and direction
of the bifurcating periodic solutions atl1are studied In Section 3, some conclusions
are stated
2 Hopf bifurcation analysis
In this section, we consider the following model:
u t = d1u + u(z1− c11u − v),
Clearly, it has one trivial equilibrium (0, 0), and three semitrivial equilibria (z1/c11,0), (0, z2/c22), and (0, z3/c22) There also has two nontrivial equilibria E1, E2:
E1=
z2− c22z1
1− c11c22
,z1− c11z2
1− c11c22
, E2=
z3− c22z1
1− c11c22
,z1− c11z3
1− c11c22
As in [4], in the following, we will limit our analysis to the case z3 >z2and z1 >c11z2,
z2>c22z1 Immediately, the condition c11c22< 1 follows as a consequence, and then E1
is a constant positive equilibrium If there has additional condition that z1 >c11z3, then
E2 is an another constant positive equilibrium E1, E2 are also positive equilibria for
Equation (2.1) without diffusion, and when E2 exists, it is unstable In fact, the linear
system at E2= (u*, v*) has the form
u t
v t
= L
u v
=
c11u∗f(c11u∗+ v∗) u∗f(c11u∗+ v∗)
v∗g(z3) c22v∗g(z3)
For f’ (c11u* + v*) = -1 and g’(z3) = z2 - z3, then the trace and determinant of L are
tr L = −c11u∗+ c22v∗(z2− z3)< 0, det L = (1 − c11c22)u∗v∗(z2− z3)< 0,
which imply that L has a positive eigenvalue, and then E2 is unstable Hence, the researchers are concerned more about the dynamics at E1 In the corresponding
diffu-sion system, the dynamics at E1 is richer than that at E2 Hence, we take our attention
to the equilibrium E1 In [4], Turing instability has been studied thoroughly The effect
on the stability due to the diffusion is analyzed In this article, we pay attention to
Hopf bifurcation bifurcated by E1 We investigate on the effect on the stability due to
the diffusion In other words, diffusion driving Hopf bifurcation is studied
Trang 3Denote l = z2- c22z1 With the conditions above, we have thatl <z3- c22z1 and 0 <
l < (1 - c11c22) z1/c11 Hence, the domain of the parameterl is 0 <l < min{z3 - c22z1,
(1 - c11c22) z1/c11} In this article, we choose l as a main bifurcation parameter and
consider the complicated dynamic behavior near the fixed point E1 with the effect of
diffusion
For convenience, we first transform the equilibrium E1 = (u*, v*) to the origin via the translation ˆu = u − λ/(1 − c11c22), ˆv = v − (z1− c11λ/(1 − c11c22)) and drop the hats
for simplicity of notation, then system (2.1) is transformed into
u t = d1u + a11u + a12v + f (u, v),
where
a11=−c11u∗, a12=−u∗, a
21 =¯zv∗, a
22= c22¯zv∗,
and
¯z =z3− u∗− c22v∗,
f (u, v) = − c11u2− uv, g(u, v) =( ¯z − 2c22v∗)uv + (c22¯z − c2
22v∗)v2
− v∗u2− u2v − 2c22uv2− c2
22v3
In the following, we consider system (2.2) on spatial domainΩ = (0, ℓπ), ℓ Î ℝ+
with Dirichlet boundary condition
u(0, t) = u( π, t) = 0, v(0, t) = v(π, t) = 0, t > 0.
Define the real-valued Sobolev space
X := {(u, v) | u, v ∈ H2(0,π), (u, v) | x=0, π = 0}, and the complexification of X by Xℂ = X + iX = {x1 + ix2|x1, x2 Î X}
The linearized operator of system (2.2) evaluated at (0, 0) is
L :=
a11+ d1∂2/∂x2 a12
a21 a22+ d2∂2/∂x2
and accordingly we define (denote μn, nÎ N are the eigenvalues of the eigenvalue problem -Δj = μj, j(0) = j(ℓπ) = 0)
L n:=
a11− d1μ n a12
a21 a22− d2μ n
Then, the characteristic equation of Ln(l) is
where
T n = a11+ a22− (d1+ d2)μ n,
D n = a11a22− a12a21− (d1a22+ d2a11)μ n + d1d2μ2
Trang 4More immediately, let Tn, Dnbe expressed by expression with parameterl:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
T n(λ) = −(d1+ d2)μ n+ c11c22
1− c11c22λ2−
2c22z1+c11c22z3− c22z1+ c11
1− c11c22
λ +c22z1(z3− c22z1),
D n(λ) = d1d2μ2
n−
d1
c11c22
1− c11c22λ2− c22d1
2z1+ c11z3− z1
1− c11c22
λ +c22d1z1(z3− c22z1)− d2
c11λ
1− c11c22
μ n+ c11
1− c11c22λ3
−
2z1+ c11z3− z1
1− c11c22
λ2+ z1(z3− c22z1)λ.
According to [5], we have Lemma 2.1 Hopf bifurcation occurs at a certain critical value l0 if there exists unique n Î N such that
T n(λ0) = 0, D n(λ0)> 0 and T j(λ0)= 0, D j(λ0)= 0 for j = n; (2:4) and for the unique pair of complex eigenvalues near the imaginary axisa(l) ± iω (l), the transversality condition a’(l0)≠ 0 holds
Let us consider the sign of Dn(l) first Denote ¯λ = min{z3− c22z1, (1− c11c22)z1/c11} Clearly, ¯λ = z3− c22z1 if c11z3 >z1 and ¯λ = (1 − c11c22)z1/c11 if c11z3 >z1 We will
prove that there exists N1Î N such that Dn(l) > 0 for all λ ∈ (0, ¯λ) and n >N1under
some simple conditions
Lemma 2.2 If z1 ≤ c11z3/2 or z1 ≥ 2c11z3, then Dn(l) > 0 for allλ ∈ (0, ¯λ)and n>N1, where N1 Î N such that μn>c22z1(z3- c22z1)/d2for n>N1
Proof First, we claim that Dn(0) > 0, D n(¯λ) > 0for all n >N1 Directly calculating,
we have
D n (0) = d1d2μ2
n − c22d1z1(z3− c22z1)μ n > 0,
D n(¯λ) =
⎧
⎨
⎩
d1d2μ2
n + d2μ n
c11(z3− c22z1)
1− c11c22 > 0 if ¯λ = z3− c22z1,
d1d2μ2
n + d2μ n z1> 0 if ¯λ = (1 − c11c22)z1/c11 Next, we prove that for all λ ∈ (0, ¯λ), Dn(l) > 0 if Dn(0) > 0, D n(¯λ) > 0 satisfied
From the expression of Dn(l), we have Dn(l) ® +∞ when l ® +∞ and Dn(l) ® - ∞
whenl ® - ∞, and Dn(l) has two inflection points for any fixed n Î N We only need
to prove that 0 and ¯λ are in the same side of the second inflection point
Differentiat-ing Dn(l) with respect to l for fixed n, we have
Dn(λ) = aλ2+ b λ + c,
where
a = 3c11
1− c11c22
,
b = −2z1− 2c11(z3− c22z1)
1− c11c22 − 2d1μ n
c11c22
1− c11c22
,
c = z1(z3− c22z1)− c22d1μ n
2z1+ c11z3− z1
1− c c
+ d2μ n
c11
1− c c .
Trang 5The axis of symmetry of Dn(λ) is
λmin= 1 3
(z3− c22z1) +1− c11c22
c11
z1+ c22d1μ n
> 0.
If z1 ≤ c11z3/2, then λmin≥ ¯λ = (1 − c11c22)z1/c11 Else if z1 ≥ 2c11z3, then
λmin≥ ¯λ = z3− c22z1 That is,0< ¯λ ≤ λmin, 0 and ¯λ are in the same side of the
sec-ond inflection point and the proof is complete
Next, we seek the critical points λ ∈ (0, ¯λ) such that Tn= 0 Define
T (λ, p) := − (d1+ d2)p + c11c22
1− c11c22λ2−
2c22z1+c11c22z3− c22z1+ c11
1− c11c22
λ + c22z1(z3− c22z1)
Then, Tn(l) = 0 is equivalent toT (λ, p) = 0 Solving p fromT (λ, p) = 0, we have
p( λ) = 1
d1+ d2
c11c22
1− c11c22λ2−
2c22z1+ c11c22z3− c22z1+ c11
1− c11c22
λ +c22z1(z3− c22z1)
Immediately,
d1+ d2
c22z1(z3− c22z1)> 0,
p(¯ λ) =
⎧
⎪
⎪
d1+ d2 ·c11(z3− c22z1)
1− c11c22 < 0 if ¯λ = z3− c22z1,
− z1
d1+ d2 < 0 if ¯λ = (1 − c11c22)z1/c11 Lemma 2.3 Denote N2∈be the number such that μ N2 ≤ p(0) < μ N2 +1 Then, there exists N2 points li, i = 1,2, , N2, satisfying ¯λ > λ1> λ2> · · · > λ N2 ≥ 0, such
that Ti(lj) < 0 for i <j, and Ti(lj) > 0 for i <j, i = 1,2, , 1≤ j ≤ N2
Lemma 2.4 Suppose li, 1≤ i ≤ N2 be defined as in Lemma 2.3 Ifa(li) ± iω(li)be the unique pair of complex eigenvalues near the imaginary axis, thena’(li) < 0
Theorem 2.5 Suppose the condition of Lemma 2.2 is satisfied and li, 1≤ i ≤ N2 be defined as in Lemma 2.3 Then, Hopf bifurcation occurs at liif
μ i < d2− d1
d1(d1+ d2)· c11λ i
1− c11c22
where N1, N2are defined as before
Proof We need to show that Dn(li) > 0, nÎ N, then Lemma 2.1 could be used First,
Ti(li) = 0 gives
(d1+ d2)μ i+ c11λ i
1− c11c22
= c11c22
1− c11c22λ2
i − c22
2z1+c11z3− z1
1− c11c22
λ i + c22z1(z3− c22z1)
Now, Dn(li) could be expressed as
D n(λ i ) =d1d2μ2
n−
d21μ i + d1d2μ i + (d1− d2) c11λ i
1− c11c22
μ n
+ c11
1− c11c22λ3
i −
2z1+c11z3− z1
1− c11c22
λ2
i + z1(z3− c22z1)λ i
Trang 6D(λ i , p) =d1d2p2−
d21μ i + d1d2μ i + (d1− d2) c11λ i
1− c11c22
p
+ c11
1− c11c22λ3
i −
2z1+ c11z3− z1
1− c11c22
λ2
i + z1(z3− c22z1)λ i
Clearly,D(λ i, 0)> 0and the axis of symmetry ofD(λ i , p)is
pmin= d
2μ i + d1d2μ i + (d1− d2)c11λ i/(1− c11c22)
2d1d2
The condition in the theorem ensure pmin < 0, which lead toD(λ i , p) > 0for p > 0
Hence, Dn(li) > 0 andliare Hopf bifurcation points
Remark2.6 Theorem 2.5 gives a sufficient condition for Hopf bifurcation occurring
From the proof of Theorem 2.5, we see that the inequality (2.5) is stringent We
con-sider thatD(λ i , p)is continuous with respect to p, but Dn(li) is a set of discrete values
Hence, we need not to ensure that the inequality (2.5) is always satisfied in some
sim-ple case For instance, N2 = 1 Example 2.8 exactly demonstrates this feature
In the following, we take attention to the stability and direction of bifurcating peri-odic solutions bifurcated atl1
We give the detail of the calculation process of the direction of Hopf bifurcation at
l1 in the following It is obvious that ±iω, with ω =D1(λ1), are the only pair of
sim-ple purely imaginary eigenvalues of L(l1) We need to calculate the Poincaré norm
form of (2.2) forl = l1:
˙z = iωz + z
M
j=1
c j (z¯z) j,
where z is a complex variable, M ≥ 1and cjare complex-valued coefficients The direction of Hopf bifurcation at l1 is decided by the sign of Re(c1), which has the
fol-lowing form:
c1= i
2ω
g20g11− 2 | g11|2−1
3 | g02|2
+1
2g21.
In the following, we will calculate g20, g11, g02, and g21 We recall that
f (u, v) = − c11u2− uv, g(u, v) =( ¯z − 2c22v∗)uv + (c22¯z − c2
22v∗)v2
− v∗u2− u2v − 2c22uv2− c2
22v3 Notice that the eigenvalues μn= n2/ℓ2
, n = 1,2, , the corresponding eigenfunction are sin(nx/ℓ) in our problem Hence, we set q = (a, b)T
sin(x/ℓ) be such that L(l1)q =
iωq and let q* = M(a*, b*)T
sin(x/ℓ) be such that L(l1)Tq* = -iωq*, and moreover, 〈q*, q〉 = 1 andq∗,¯q = 0 Here
u, v =
π
0 ¯u T vdx, u, v ∈ X
Trang 7be the inner dot and
a = b∗= 1, b = iω + d1μ1− a11
a12
, a∗= −iω + d2μ1− a22
a12
ia12 Express the partial derivatives of f(u, v) and g(u, v) at (u, v) = (0, 0) with respect tol whenl1, we have
f uu=−c11, f uv=−1, g uv = z3− 3c22z1+λ1(3c11c22− 1)
1− c11c22
,
g uu=−z1+ c11λ1
1− c11c22
, g vv = c22(z3− 2c22z1) +c22λ1(2c11c22− 1)
1− c11c22
,
g vvv=−c2
22, g uuv=−1, g uvv=−2c22, and the others are equal to zero As stated in [5,6], we need to calculate Q qq , Q q ¯q, and C qq ¯q, which are defined as
Q qq= sin2(x/ )
c d
, Q q ¯q= sin2(x/ )
e f
, C qq ¯q= sin3(x/ )
g h
,
where
⎧
⎪
⎪
c = f uu a2+ 2f uv ab + f vv b2, d = g uu a2+ 2g uv ab + g vv b2,
e = f uu | a|2+ f uv (a¯b + ¯ab) + f vv | b|2, f = g uu | a|2+ g uv (a¯b + ¯ab) + g vv | b|2,
g = f uuu | a|2a + f uuv(2| a|2b + a2¯b) + f uvv(2| b|2a + b2¯a) + f vvv | b|2b,
h = g uuu | a|2a + g uuv(2| a|2b + a2¯b) + g uvv(2| b|2a + b2¯a) + g vvv | b|2b.
From direct calculation, we have
q∗, Q
qq =4 ¯M
3 (¯a∗c + d), q∗, Q
q ¯q =4 ¯M
3 (¯a∗e + f ),
¯q∗, Q
qq =4M
3 (a
∗c + d), ¯q∗, Q
q ¯q =4M
3 (a
Then, we have (the detail meaning of the following parameters are stated in [6,5])
H20= Q qq − q∗, Q
qq q − ¯q∗, Q
qq ¯q
= 1
2(1− cos(2x/))
c d
−
q∗, Q
a b
− ¯q∗, Q
¯a
¯b
sin(x/ )
=
∞
k=1
−8
(2k − 1)(2k + 1)(2k − 3)π
c d
sin((2k − 1)x/)
−
q∗, Q
1
b
− ¯q∗, Q
1
¯b
sin(x/ )
(2:7)
and
H11= Q q ¯q − q∗, Q
q ¯q q − ¯q∗, Q
q ¯q ¯q
= 1
2(1− cos(2x/))
e f
−
q∗, Q
q ¯q
a b
− ¯q∗, Q
q ¯q
¯a
¯b
sin(x/ )
=
∞
k=1
−8
(2k − 1)(2k + 1)(2k − 3)π
e f
sin((2k − 1)x/)
−
q∗, Q
q ¯q
1
b
− ¯q∗, Q
q ¯q
1
¯b
sin(x/ ).
(2:8)
Therefore, we can obtain w20, w11as
w20 = [2i ωI − L(λ1)]−1H20 and w11=−[L(λ1)]−1H11
Trang 8Clearly, the calculation of (2iωI - L(l1))-1and [L(l1)]-1are restricted to the subspaces spanned by the eigenmodes sin(kx/ℓ), k = 1,2, One can compute that
(2i ωI − L k(λ1))−1
= (α k
1+ i α k
2)−1
2i ω − a22+ d2μ k a12
a21 2i ω − a11+ d1μ k
,
L−1k (λ1) = 1
α k
3
a22− d2μ k −a12
−a21 a11− d1μ k
, where
α k
1=−4ω2+ a11a22− a12a21− (d1a22+ d2a11)μ k + d1d2μ2
k,
α k
2=−2ω(a11+ a22) + 2ω(d1+ d2)μ k,
α k
3= a11a22− a12a21− (d2a11+ d1a22)μ k + d1d2μ2
k Then,
w20=
∞
k=1
−8 sin((2k − 1)x/) (2k − 1)(2k + 1)(2k − 3)π (2i ωI − L 2k−1(λ1 ))−1
c d
− (2iωI − L1 (λ1 ))−1
q∗, Q
a b
− ¯q∗, Q
¯a
¯b
sin(x/ )
=
∞
k=1
−8 sin((2k − 1)x/)(2k − 3)−1
(4k2− 1)(α 2k−1
1 + i α 2k−1
(2i ω − a22+ d2μ 2k−1 )c + a12d
a21c + (2i ω − a11+ d1μ 2k−1 )d
− 1
α1+ i α1
(2i ω − a22+ d2μ1 )ξ1+ a12ξ2
a21ξ1+ (2i ω − a11+ d1μ1 )ξ2
sin(x/ ),
w11=
∞
k=1
−8 sin((2k − 1)x/)
α 2k−13 (4k2− 1)(2k − 3)π −
(a22− d2μ 2k−1 )e + a12f
a21e − (a11− d1μ 2k−1 )f
−α11
−(a22− d2μ1 )ξ3+ a12ξ4
a21ξ3− (a11− d1μ1 )ξ4
sin(x/ ),
where
ξ1=q∗, Q
qq a − ¯q∗, Q
qq ¯a = 4c
3 (¯a∗¯M − a∗M) + 4d
3 ( ¯M − M),
ξ2=q∗, Q
qq b − ¯q∗, Q
qq ¯b = 4c
3 (b¯a∗¯M − ¯ba∗M) + 4d
3 (b ¯ M − ¯bM),
ξ3=q∗, Q
q ¯q a − ¯q∗, Q
q ¯q ¯a = 4e
3 (¯a∗¯M − a∗M) + 4f
3 ( ¯M − M),
ξ4=q∗, Q
q ¯q b − ¯q∗, Q
q ¯q ¯b = 4e
3 (b ¯a∗ ¯M − ¯ba∗M) + 4f
3 (b ¯ M − ¯bM).
Then,
Q w20¯q=
∞
k=1
Q 1k
w20¯q
Q 2k
w20¯q
sinx
sin
(2k − 1)x
Q10
w20¯q
Q20
w20¯q
sin2x
=
∞
k=1
f uu w 1k
20+ f uv ¯bw 1k
20+ f uv w 2k
20
g uu w 1k
20+ g uv ¯bw 1k
20+ g uv w 2k
20+ g vv ¯bw 2k
20
sinx
sin
(2k − 1)x
+
f uu w10
20+ f uv ¯bw10
20+ f uv w20
g uu w1020+ g uv ¯bw10
20+ g uv w20+ g vv ¯bw20
sin2x
,
Q w11q=
∞
k=1
Q 1k
w11q
Q 2k
w11q
sinx
sin
(2k − 1)x
Q10
w11q
Q20
w11q
sin 2x
,
=
∞
k=1
f uu w 1k
11+ f uv bw 1k
11+ f uv w 2k
11
g uu w 1k11+ g uv bw 1k11+ g uv w 2k11+ g vv bw 2k11
sinx
sin
(2k − 1)x
+
f uu w10
11+ f uv bw10
11+ f uv w20 11
g w10+ g bw10+ g w20+ g bw20
sin2x
,
Trang 9w 1k20 =
∞
k=1
−8(2iω − a22+ d2μ 2k−1)c + a12d) (4k2− 1)(2k − 3)(α 2k−1
1 + i α 2k−1
2 )π, k = 1, 2, ,
w 2k20 =
∞
k=1
−8(a21c + (2iω − a11+ d1μ 2k−1 )d) (4k2− 1)(2k − 3)(α 2k−1
1 + i α 2k−1
2 )π, k = 1, 2, ,
w 1k11 =
∞
k=1
−8(−(a22− d2μ 2k−1 )e + a12f )
α 2k−1
3 (4k2− 1)(2k − 3)π , k = 1, 2, ,
w 2k11 =
∞
k=1
−8(a21e − (a11− d1μ 2k−1)f )
α32k−1 (4k2− 1)(2k − 3)π , k = 1, 2, ,
and
w1020= (2i ω − a22+ d2μ1)ξ1+ a12ξ2
α1
1+ i α1 2
, w2020= a21ξ1+ (2i ω − a11+ d1μ1)ξ2
α1
1+ i α1 2
,
w1011= −(a22− d2μ1)ξ3+ a12ξ4
α1 , w2011 = a21ξ3− (a11− d1μ1)ξ4
Notice that
π 0
sin4(x/ )dx =3π
8 ,
π 0
sin2(x/ ) sin((2k − 1)x/)dx = −4
(2k − 1)(2k + 1)(2k − 3),
we have
q∗, C
qq ¯q = 3 ¯Mhπ
q∗, Q
w20¯q =
∞
k=1
−4 ¯M (2k − 1)(2k + 1)(2k − 3)(¯a∗Q 1k w20¯q + Q 2k w20¯q)
+4M
3 (¯a∗Q10w20¯q + Q20w20¯q),
q∗, Q
w11q =∞
k=1
−4 ¯M (2k − 1)(2k + 1)(2k − 3)(¯a∗Q 1k w11q + Q 2k w11q) +4M
3 (¯a∗Q10
w11¯q + Q20w11¯q).
Hence, we have
g20=q∗, Q
qq = 4 ¯M
3 (¯a∗c + d),
g11=q∗, Q q ¯q = 4 ¯M
3 (¯a∗e + f ),
g02=q∗, Q ¯qq = 4 ¯M
3 (¯a∗¯c + ¯d),
Trang 10g21= 2q∗, Q
w11q + q∗, Q
w20¯q + q∗, C
qq ¯q
=
∞
k=1
−4 ¯M((2Q 1k
w11q + Q 1k w
20¯q)¯a∗
n + (2Q 2k w11q + Q 2k w
20¯q)) (2k − 1)(2k + 1)(2k − 3)
+4 ¯M((2Q10
w11q + Q10w
20¯q)¯a∗+ (2Q20
w11q + Q20w
20¯q))
3 ¯Mhπ
Then, it follows that
c1= i
2ω (g20g11− 2|g11|2−1
3|g02|2) +1
2g21
= 82i
9ω[ ¯M2(¯a∗c + d)( ¯a∗e + f ) − 2|M|2|¯a∗e + f|2−
1
3|M|2|¯a∗c + d|2] +
∞
k=1
−2 ¯M((2Q 1k
w11q + Q 1k w
20¯q)¯a∗
n + (2Q 2k
w11q + Q 2k w
20¯q)) (2k − 1)(2k + 1)(2k − 3)
+2 ¯M((2Q10
w11q + Q10w
20¯q)¯a∗+ (2Q20
w11q + Q20w
20¯q))
3 ¯Mhπ
Theorem 2.7 Suppose the conditions in Theorem 2.7 are satisfied Then, the positive constant equilibrium E1 is asymptotically stable when λ ∈ (λ1, ¯λ) Hopf bifurcation
occurs at l1, and the bifurcating periodic solutions are in the left(right) neighborhood of
l1 and stable(unstable) if Re(c1) < 0(> 0)
Example 2.8 Suppose ℓ = 1(i e Ω = (0, π)) d1 = 1/10, d2 = 3/10, z1 = z2= 1, z3 = 3/2 and c11 = 1/3 Let c22be the bifurcation parameter We found that there has only
one Hopf bifurcation point l = 0.0833 E1 is stable for 0.0833 <l < 1.1667 For l <
0.0833, Hopf bifurcation occurs and the bifurcating periodic solutions are stable In
other words, c22= 0.9167 is the critical value for Hopf bifurcation We give the
simula-tion for c220.9167 ± 0.02 in the follows If c22= 0.9167 - 0.02, E1is stable (Figure 1) If
c22= 0.9167 + 0.02, there exists periodic solution, which is stable (Figure 2)
3 Conclusion
In this article, we take l as a main bifurcation parameter, study stability of the
con-stant positive equilibrium E1, which exists for λ ∈ (0, ¯λ) The critical values for Hopf
0
2
4 0
100 200
3000 0.5 1
Distance x
u(x,t)
2
4 0
100 200
3000 0.5 1 1.5
Distance x
v(x,t)
Time t Figure 1 E is asymptotically stable for c = 0.9167 - 0.02 The initial value is (u , v ) = (0.1, 0.2) * sin x.
... w11=−[L(λ1)]−1H11 Trang 8Clearly, the calculation of (2iωI...
0 ¯u T vdx, u, v ∈ X
Trang 7be the inner dot and< /p>
a =...
− (a< /small>22− d2μ1 )ξ3+ a< /small>12ξ4